We examine various scenarios in which the Standard Model is extended by a light leptoquark state to explain one or both B-physics anomalies. Combining low-energy constraints and direct searches at the LHC, we confirm that the only single leptoquark model that can explain both anomalies at the same time is a vector leptoquark, known as U1. Focusing on U1, we highlight the complementarity between LHC and low-energy constraints, and argue that improving the experimental bound on B(B → Kµτ ) by two orders of magnitude could compromise its viability as a solution to the B-physics anomalies.